I can't help with this problem, which probably requires a fairly extensive knowledge of semidirect products (and in any case I am about to go away for a week). But the problem as stated has two obvious mistakes.

First, the definition of is wrong. It does not define an automorphism group. In fact, the definition of does not even define a continuous function of t at integer values of . The correct definition should presumably be when .

Second, the semidirect product does not make sense. It should be